# Can a log have multiple bases?

• B
• YouAreAwesome
In summary: That's right. The exam is an Australian NSW HSC exam created by the school. Binary/hexadecimal etc is not included in the NSW syllabus and therefore can not be examined or tested. ##(\log_e)_2## means "log to the base e to the base 2" where "to the base e" and "to the base 2" have identical meanings with regard to "to the base".##(\log_e)_2## means "log to the base e to the base 2" where "to the base e" and "to the base 2" have identical meanings with regards to "to the base".
If I follow the semantics correctly, we start with the well-accepted: ##\log_b x = \frac{\log x}{\log b}##
Equally well accepted is: ##\ln x = \log_e x = \frac{\log x}{\log e}##

I see two possible ways to systematically extend this notation scheme to supply a meaning for ##\ln_b x##.

Suppose that we define the meaning of any subcripted function ##f_b(x)## to be ##\frac{f(x)}{f(b)}##
In particular, ##\ln_2 x = \frac{\ln x}{\ln 2} = \frac {\frac{\log x}{\log e}} {\frac{\log 2}{\log e}} = \frac{\log x}{\log 2} = \log_2 x##

So this possible definition would mean that there is no such thing as a log with two bases. Only logs with one base. Any subscript renders the original base irrelevant.

Suppose, alternately, that we define the meaning of any subscripted function ##f_b(x)## to be ##\frac{f(x)}{\log b}##
In particular, ##\ln_2 x = \frac{\ln x}{\log 2} = \frac {\frac{\log x}{\log e}} {\log 2} = \frac {\log x}{\log 2e} = \log_{2e}x##
Apparently this was the selected meaning. Ludicrous.

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Nik_2213, Merlin3189, YouAreAwesome and 1 other person
YouAreAwesome said:
Has anyone ever seen this notation before?
No, it's nonsense. What exam board/country is this in? Is there a formally published syllabus?

YouAreAwesome, Mark44 and fresh_42
pbuk said:
No, it's nonsense. What exam board/country is this in? Is there a formally published syllabus?
Australia, NSW Stage 6 Mathematics Syllabus.

Outcome:

MA11-6 manipulates and solves expressions using the logarithmic and index laws, and uses logarithms and exponential functions to solve practical problems

Pages 43-44 outline the course "logarithm" content, and page 54 outlines the course "derivative of natural logarithm" content.

I'm not sure which company created the exam itself. Exams are generally purchased from either CSSA, Independent, or ACE.

pbuk
Seems like a thinly veiled attempt to avoid paying Australia's fair share of the usage fee for the common (Briggsian) logarithm. The royalty checks have been running light lately.

[Very light. I've never seen a dime of that money]

hutchphd, YouAreAwesome and fresh_42
Well that notation is not used anywhere in the syllabus you linked, nor in a couple of past papers I looked at from the same site, nor in the sample teaching plan https://educationstandards.nsw.edu.au/wps/portal/nesa/11-12/resources/sample-units

I would guess that what we have here is a Head of Department that has lost (or never had) his abilities and staff that can't upset the apple cart for fear of their job. That puts you, and your unfortunate tutee, in a very weak position.

Is the mark from this internal exam significant?

Mark44, YouAreAwesome, DrClaude and 1 other person
pbuk said:
Well that notation is not used anywhere in the syllabus you linked, nor in a couple of past papers I looked at from the same site, nor in the sample teaching plan https://educationstandards.nsw.edu.au/wps/portal/nesa/11-12/resources/sample-units

I would guess that what we have here is a Head of Department that has lost (or never had) his abilities and staff that can't upset the apple cart for fear of their job. That puts you, and your unfortunate tutee, in a very weak position.

Is the mark from this internal exam significant?
How significant is an exam that assesses your mathematical knowledge gained over the past 13 years of schooling and enables you access into university courses? I would think the exam is one of the most significant exams a person will face in their entire lives.

But to be more specific, I believe it contributes ~30% of the School Based mark. The school based mark ranks the students is worth 50% of their Higher School Certificate mark from memory.

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YouAreAwesome said:
How significant is an exam that assesses your mathematical knowledge gained over the past 13 years of schooling and enables you access into university courses?
I didn't know that in NSW (and presumably Australia generally) that that is the case for internally set exams. In England for instance the only results that count are the public exams set nationally where each paper is taken by tens of thousands of students.

If such an error were to slip through the exam preparation it would not be up to the candidates to take it up with their schools, it would be the schools themselves taking it up with the exam board.

There must be a similar appeal process in NSW: I suggest you work with your tutee and their parents or guardians to follow this. The first step would be to determine if not receiving a mark for the erroneous question affected any final grading (bearing in mind that raw marks are usually moderated or weighted to produce a final mark).

I don't know if there is anyone on PF that is close to the NSW secondary education system, you could try posting in https://www.physicsforums.com/forums/stem-educators-and-teaching.192/ referring to this thread, otherwise I suggest you look locally for some independent body that can guide them through the process.

YouAreAwesome
pbuk said:
I didn't know that in NSW (and presumably Australia generally) that that is the case for internally set exams. In England for instance the only results that count are the public exams set nationally where each paper is taken by tens of thousands of students.

If such an error were to slip through the exam preparation it would not be up to the candidates to take it up with their schools, it would be the schools themselves taking it up with the exam board.

There must be a similar appeal process in NSW: I suggest you work with your tutee and their parents or guardians to follow this. The first step would be to determine if not receiving a mark for the erroneous question affected any final grading (bearing in mind that raw marks are usually moderated or weighted to produce a final mark).

I don't know if there is anyone on PF that is close to the NSW secondary education system, you could try posting in https://www.physicsforums.com/forums/stem-educators-and-teaching.192/ referring to this thread, otherwise I suggest you look locally for some independent body that can guide them through the process.
Thanks for your response. NSW school based exams determine ranking within the school for that particular course. The mark the teachers send to the NSW Education Standards Authority (NESA) is worth 50% of the Higher School Certificate grade for that course. The other 50% is counted from an external exam.

In this case, all the students in the class were penalised in the same way by being presented with a ridiculous notation that is incoherent. If the school were to take this further the whole class would benefit by removing the total possible marks by two and this would mean the mark the teacher sends to NESA would be slightly higher.

The argument against following this up is that all students gain equally. I do not take this approach as teachers should be assessing students against outcomes so that the marks they send to NESA are significant even without any internal ranking system.

But is it worth my effort and time to take this further? I'm not quite sure...

YouAreAwesome said:
But is it worth my effort and time to take this further?
No.

A teacher had a strange idea and told it.
A student realized it wasn't actually a good idea and asked.
The authorities did what authorities always do: they invented a justification to keep the authority.

From that point on, it isn't about the idea anymore, and things can only get worse. For all of you.

My advice: Forget the teacher's idea and forget this storm in a glass of water.

YouAreAwesome and jbriggs444
fresh_42 said:
No.

A teacher had a strange idea and told it.
A student realized it wasn't actually a good idea and asked.
The authorities did what authorities always do: they invented a justification to keep the authority.

From that point on, it isn't about the idea anymore, and things can only get worse. For all of you.

My advice: Forget the teacher's idea and forget this storm in a glass of water.
Noted.

pbuk

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